On-Line Electronic Supplementary Material A: Supporting Discussions
Instrumental Variables
The inverse demand function in an imperfectly competitive market depends on the output levels
of all firms. However, in some cases other factors influence the price. Three cases are discussed
below. In the body of the paper we assume the simplest case, (1) estimating the inverse demand
function with no omitted variables, 𝑃𝑖 = 𝑃0 − 𝛼𝒴𝑖 + 𝜀𝑖 , and 𝐸(𝒴𝑖 𝜀𝑖 ) = 0, 𝜀𝑖 ~𝑁(0, 𝜎𝜀2 ) i.i.d. and
a regression using Ordinary Least Squares (OLS) and 𝐸(𝑃𝑖 |𝒴𝑖 = 𝑌𝑖 ) = 𝑃0 − 𝛼𝑌𝑖 . However, if
there exist omitted variables 𝑊𝑖 which affect price 𝑃𝑖 , such that 𝑃𝑖 = 𝑃0 − 𝛼𝒴𝑖 + 𝛽𝑊𝑖 + 𝜂𝑖 ,
where 𝜂𝑖 ~𝑁(0, 𝜎𝜂2 ), alternative cases need to be considered.1 In case (2), OLS can still provide
consistent estimates when 𝐸(𝒴𝑖 𝜂𝑖 ) = 0 , 𝐸(𝑊𝑖 𝜂𝑖 ) = 0 and the quantity variable 𝒴𝑖 is
uncorrelated with the omitted variable, i.e., 𝐸(𝒴𝑖 𝑊𝑖 ) = 0 . Thus, the regression generates
0
0
𝐸(𝑃𝑖 |𝒴𝑖 = 𝑌𝑖 ) = 𝑃𝑊
− 𝛼𝑌𝑖 , where 𝑃𝑊
= 𝑃0 + 𝛽𝐸(𝑊𝑖 ). In case (3), if 𝒴𝑖 is correlated with the
omitted variable 𝐸(𝒴𝑖 𝑊𝑖 ) ≠ 0, let 𝜀𝑖 = 𝛽𝑊𝑖 + 𝜂𝑖 , which results in 𝐸(𝒴𝑖 𝜀𝑖 ) ≠ 0.
Case (3) is termed endogeneity in econometrics; OLS provides inconsistent, biased, and
inefficient estimates for the 𝛼 parameters of interest (Greene, 2011). To address this issue, we use
an instrumental variable 𝑍𝑖 that is highly correlated with 𝒴𝑖 but independent of 𝑊𝑖 and 𝜂𝑖 ,
specifically 𝐸(𝑍𝑖 𝑊𝑖 ) = 0, 𝐸(𝑍𝑖 𝜂𝑖 ) = 0. The linear regression model can be rewritten as 𝑃𝑖 =
𝑃0 − 𝛼𝑍𝑖 + 𝛽𝑊𝑖 + 𝜂𝑖 , and OLS can provide consistent estimates and the expected value
0
𝐸(𝑃𝑖 |𝑍𝑖 = 𝑧𝑖 ) = 𝑃𝑊
− 𝛼𝑧𝑖 . As described in Section 3.1, our paper focuses on an inverse demand
function expressed and estimated by a linear function 𝑃(𝑌) = 𝑃0 − 𝛼𝑌, where 𝑃0 is the intercept
corresponding to case 1. However, if endogeneity exists in the inverse demand model, we can
identify instrumental variables using the methods described in Goldberger (1972), Morgan
(1990), and Angrist and Krueger (2001). Note that, when output quantity changes, we assume a
change in supply curve rather than a change in quantity supplied.
The omitted variable could be the price or quantity of substitute products or other contextual factors that could
affect the price of output q.
1
Weak, Moderate, and Strong Dominance Properties
Lemma A1: In the two-output product case, if matrix 𝜶 satisfies MDD and symmetric
properties, then matrix 𝜶 satisfies SDD.
Proof: If matrix 𝜶 satisfies MDD and symmetric properties, then the transitivity property follows
(i.e., If 𝛼𝑞𝑞 ≥ 𝛼𝑞ℎ and 𝛼𝑞ℎ ≥ 𝛼ℎ𝑞 , then 𝛼𝑞𝑞 ≥ 𝛼ℎ𝑞 , for all 𝑞 ≠ ℎ), which implies that the main
effect of each product dominates the minor effect of the other products, i.e., the SDD property.
□
Lemma A2: If price sensitivity matrix 𝜶 satisfies the SDD property, then solving MiCP (5)
̂ 𝑞 ) ≥ 0, where ∀(𝑋𝑟𝑖 , 𝑦𝑟𝑞 ) ∈ 𝑇̃.
generates a solution such that 𝑦𝑟𝑞 ≥ 0 and 𝑃𝑞 (𝑌𝑞 , 𝒀
̂ 𝑞 ) − 𝛼𝑞𝑞 𝑦𝑟𝑞 − ∑ℎ≠𝑞 𝛼ℎ𝑞 𝑦𝑟ℎ − 𝜇1𝑟𝑞 = 0 , we can obtain the inequality
Proof: Let 𝑃𝑞 (𝑌𝑞 , 𝒀
̂ 𝑞 ) ≥ 𝛼𝑞𝑞 𝑦𝑟𝑞 + ∑ℎ≠𝑞 𝛼ℎ𝑞 𝑦𝑟ℎ . We enumerate all combinations of 𝑦𝑟𝑞 and 𝑦𝑟ℎ being either
𝑃𝑞 (𝑌𝑞 , 𝒀
̂ 𝑞 ) ≥ 0 and the revenue function is
non-negative or positive. If 𝑦𝑟𝑞 , 𝑦𝑟ℎ ≥ 0, 𝑞 ≠ ℎ, then 𝑃𝑞 (𝑌𝑞 , 𝒀
non-negative. The case 𝑦𝑟𝑞 ≥ 0, 𝑦𝑟ℎ < 0, 𝑞 ≠ ℎ will not happen because, given function
̂ 𝑞 ) = 𝑃𝑞0 − 𝛼𝑞𝑞 𝑌𝑞 − ∑ℎ≠𝑞 𝛼𝑞ℎ 𝑌ℎ , the increase in revenue due to the increase in price
𝑃𝑞 (𝑌𝑞 , 𝒀
̂ 𝑞 ) ≥ 0 cannot exceed the decrease of revenue through the increase of price
𝑃𝑞 (𝑌𝑞 , 𝒀
̂ ℎ ) ≥ 0. Price 𝑃𝑞 (𝑌𝑞 , 𝒀
̂ 𝑞 ) is not sensitive with respect to 𝑦𝑟ℎ due to the symmetry of 𝜶
𝑃ℎ (𝑌ℎ , 𝒀
and 𝛼𝑞𝑞 ≫ 𝛼𝑞ℎ for all 𝑞, as 𝑦𝑟ℎ becomes more negative and 𝑦𝑟ℎ < 0. Thus, the solution for
revenue maximization problem is 𝑦𝑟ℎ = 0. Similarly 𝑦𝑟𝑞 < 0, 𝑦𝑟ℎ ≥ 0 for all 𝑞 ≠ ℎ, by the
̂ 𝑞 ) ≥ 0, which will cause 𝑦𝑟𝑞 = 0 to maximize revenue. If 𝑦𝑟𝑞 <
SDD property we have 𝑃𝑞 (𝑌𝑞 , 𝒀
̂ 𝑞 ) > 0. Similar to Lemma 3.1, we have solution 𝑦𝑟𝑞 =
0, 𝑦𝑟ℎ < 0, 𝑞 ≠ ℎ, then we have 𝑃𝑞 (𝑌𝑞 , 𝒀
̂ 𝑞 )𝑦𝑟𝑞 > 0 . Therefore, the case 𝑦𝑟𝑞 <
𝑦𝑟ℎ = 0 to maximize the revenue function ∑𝑞 𝑃𝑞 (𝑌𝑞 , 𝒀
0, 𝑦𝑟ℎ < 0 will not happen. Moreover, if price sensitivity matrix 𝜶 is symmetric and satisfies the
WDD property, and 𝛼𝑞𝑞 ≫ 𝛼𝑞ℎ for all q, then, as shown in Lemma A1, solving MiCP (5) will
̂ 𝑞 ) ≥ 0, where ∀(𝑋𝑟𝑖 , 𝑦𝑟𝑞 ) ∈ 𝑇̃
automatically generate 𝑦𝑟𝑞 ≥ 0 and 𝑃𝑞 (𝑌, 𝒀
□
Lemma A2 shows a case of SDD of sensitivity matrix 𝜶. The WDD or MDD properties
are not enough to ensure 𝑦𝑟𝑞 ≥ 0 in MiCP (5). That is, if 𝜶 is not symmetric or violates MDD,
then, for some 𝑦𝑟𝑞 , a Nash equilibrium solution may set 𝑦𝑟𝑞 < 0. We illustrate this in two cases
below.
Case 1: If price sensitivity matrix 𝜶 satisfies MDD but not symmetry, and given 𝑦𝑟𝑞 ≠ 0 ,
̂ 𝑞 ) − 𝛼𝑞𝑞 𝑦𝑟𝑞 − ∑ℎ≠𝑞 𝛼ℎ𝑞 𝑦𝑟ℎ − 𝜇1𝑟𝑞 = 0
𝑃𝑞 (𝑌𝑞 , 𝒀
then
̂ 𝑞 )−∑ℎ≠𝑞 𝛼ℎ𝑞 𝑦𝑟ℎ −𝜇1𝑟𝑞
𝑃𝑞 (𝑌𝑞 ,𝒀
𝛼𝑞𝑞
∑𝑘≠𝑟 𝑦𝑘𝑞
2
=
.
𝑃𝑞0 −𝛼𝑞𝑞 𝑌𝑞 −∑ℎ≠𝑞 𝛼𝑞ℎ 𝑌ℎ −∑ℎ≠𝑞 𝛼ℎ𝑞 𝑦𝑟ℎ −𝜇1𝑟𝑞
𝛼𝑞𝑞
𝛼
𝛼
𝜇1𝑟𝑞
− ∑ℎ≠𝑞 2𝛼𝑞ℎ 𝑌ℎ − ∑ℎ≠𝑞 2𝛼ℎ𝑞 𝑦𝑟ℎ − 2𝛼
𝑞𝑞
𝑞𝑞
We
have
𝑦𝑟𝑞 =
𝑃𝑞0
. Finally, we have 𝑦𝑟𝑞 = 2𝛼 −
𝑞𝑞
. Thus, 𝑦𝑟𝑞 might be less than zero and
𝑞𝑞
̂ 𝑞 ) < 0 for the revenue maximization problem as 𝛼ℎ𝑞 ≫ 𝛼𝑞𝑞 and 𝑦𝑟ℎ > 0.
𝑃𝑞 (𝑌𝑞 , 𝒀
Case 2: If price sensitivity matrix 𝜶 satisfies WDD and symmetric properties, in an extreme case:
𝛼
for any one product 𝑞, then 𝛼𝑞ℎ → 1− , where this notation means the ratio approaches 1 from the
𝑞𝑞
left-hand side. We know 𝑦𝑟𝑞 =
𝑃0
Then, 𝑦𝑟𝑞 ≈ 2𝛼𝑞 −
∑𝑘≠𝑟 𝑦𝑘𝑞
2
𝑞𝑞
−
̂ 𝑞 )−∑ℎ≠𝑞 𝛼ℎ𝑞 𝑦𝑟ℎ −𝜇1𝑟𝑞
𝑃𝑞 (𝑌𝑞 ,𝒀
𝛼𝑞𝑞
∑ℎ≠𝑞 𝑌ℎ
2
−
∑ℎ≠𝑞 𝑦𝑟ℎ
2
=
𝑃𝑞0 −𝛼𝑞𝑞 𝑌𝑞 −∑ℎ≠𝑞 𝛼𝑞ℎ 𝑌ℎ −∑ℎ≠𝑞 𝛼ℎ𝑞 𝑦𝑟ℎ −𝜇1𝑟𝑞
𝛼𝑞𝑞
.
𝜇1𝑟𝑞
− 2𝛼 . Thus, 𝑦𝑟𝑞 might be less than zero and
𝑞𝑞
̂ 𝑞 ) > 0 for the revenue maximization problem since 𝛼𝑞𝑞 and 𝛼ℎℎ are large, and
𝑃𝑞 (𝑌𝑞 , 𝒀
𝛼𝑞𝑞 > 𝛼ℎℎ and 𝑦𝑟ℎ > 0.
Therefore, to ensure 𝑦𝑟𝑞 ≥ 0 from formulation (5), the SDD property provides a
sufficient condition based on Lemma A2. If matrix 𝜶 satisfies SDD and given an extreme case:
for all 𝑞,
𝑠𝑢𝑚(𝜶)−𝑡𝑟(𝜶)
𝛼𝑞𝑞
If we know 𝑦𝑟𝑞 =
→ 0+ , this notation means the ratio approaches 0 from the right-hand side.
̂ 𝑞 )−∑ℎ≠𝑞 𝛼ℎ𝑞 𝑦𝑟ℎ −𝜇1𝑟𝑞
𝑃𝑞 (𝑌𝑞 ,𝒀
𝛼𝑞𝑞
𝑃𝑞0 −𝛼𝑞𝑞 ∑𝑘≠𝑟 𝑦𝑘𝑞 −∑ℎ≠𝑞 𝛼𝑞ℎ 𝑌ℎ −∑ℎ≠𝑞 𝛼ℎ𝑞 𝑦𝑟ℎ −𝜇1𝑟𝑞
2𝛼𝑞𝑞
maximization problem.
≈
, then we can obtain an estimate of 𝑦𝑟𝑞 =
𝑃𝑞0 −𝛼𝑞𝑞 ∑𝑘≠𝑟 𝑦𝑘𝑞 −𝜇1𝑟𝑞
2𝛼𝑞𝑞
>0
for
the
revenue
Cost minimization case
In the case of a single fixed input and a single variable input and a given output level, Fig.A1
illustrates the Nash equilibrium solution obtained by minimizing costs. Each firm attempts to
adjust its variable input to reach the isoquant, holding a fixed input constant in the short run.
Fig.A1 Adjusted variable input in Nash equilibrium
We construct a multi-input cost model to identify a Nash equilibrium solution using MiCP. The
result shows that the Nash equilibrium solution is on the production frontier, regardless of the 𝜷
matrix selected. In particular, to formulate the MiCP with multiple variable inputs, first we define
the Lagrangian function as:
𝑉
𝑉
𝑉
̂𝑗𝑉 )𝑥𝑟𝑗
𝐿𝑟 (𝑥𝑟𝑗
, 𝜆𝑟𝑘 , 𝜇1𝑟 , 𝜇2𝑟𝑖 , 𝜇3𝑟𝑗 , 𝜇4𝑟 ) ∶= − ∑𝑗 𝑃𝑗𝑋 (𝑋𝑗𝑉 , 𝑿
− ∑𝑞 𝜇1𝑟𝑞 (𝑌𝑟𝑞 − ∑𝑘 𝜆𝑟𝑘 𝑌𝑘𝑞 ) −
𝑉
𝑉
𝐹
𝐹
∑𝑖 𝜇2𝑟𝑖 (∑𝑘 𝜆𝑟𝑘 𝑋𝑘𝑖 − 𝑋𝑟𝑖 ) − ∑𝑗 𝜇3𝑟𝑗 (∑𝑘 𝜆𝑟𝑘 𝑋𝑘𝑗 − 𝑥𝑟𝑗 ) − 𝜇4𝑟 (∑𝑘 𝜆𝑟𝑘 − 1).
Then, the resulting MiCP problem is:
𝑉
𝜕𝐿
𝑉
𝑉
𝑉
̂𝑗𝑉 ) − 𝛽𝑗𝑗 𝑥𝑟𝑗
0 ≥ 𝜕𝑥 𝑉𝑟 = (−𝑃𝑗𝑋 (𝑋𝑗𝑉 , 𝑿
− ∑𝑙≠𝑗 𝛽𝑙𝑗 𝑥𝑟𝑙
+ 𝜇3𝑟𝑗 ) ⊥ 𝑥𝑟𝑗
≥ 0, ∀𝑟, 𝑗
𝑟𝑗
𝜕𝐿𝑟
0 ≥ 𝜕𝜆
𝑟𝑘
𝑉
𝐹
= (∑𝑞 𝜇1𝑟𝑞 𝑌𝑘𝑞 − ∑𝑖 𝜇2𝑟𝑖 𝑋𝑘𝑖
− ∑𝑗 𝜇3𝑟𝑗 𝑋𝑘𝑗
− 𝜇4𝑟 ) ⊥ 𝜆𝑟𝑘 ≥ 0, ∀𝑟, 𝑘
0 ≥ (𝑌𝑟𝑞 − ∑𝑘 𝜆𝑟𝑘 𝑌𝑘𝑞 ) ⊥ 𝜇1𝑟𝑞 ≥ 0, ∀𝑟, 𝑞
𝐹
𝐹
) ⊥ 𝜇2𝑟𝑖 ≥ 0, ∀𝑟, 𝑖
0 ≥ (∑𝑘 𝜆𝑟𝑘 𝑋𝑘𝑖
− 𝑋𝑟𝑖
𝑉
𝑉
0 ≥ (∑𝑘 𝜆𝑟𝑘 𝑋𝑘𝑗 − 𝑥𝑟𝑗 ) ⊥ 𝜇3𝑟𝑗 ≥ 0, ∀𝑟, 𝑗
0 = (∑𝑘 𝜆𝑟𝑘 − 1), ∀𝑟.
(A1)
Theorem A1: In the cost minimization case, a Nash equilibrium generated from MiCP (A1)
exists on the production frontier, given an arbitrary 𝜷 matrix with all non-negative components
satisfying WDD.
Proof: Proving the existence of a Nash equilibrium is similar to Theorem 4.2. The Nash
equilibrium generated from MiCP will stay on the production frontier, given an arbitrary 𝜷
𝑉
matrix satisfying WDD. If an equilibrium output vector exists and 𝑥𝑟𝑗
> 0 , then it must satisfy
the first order condition of MiCP as the complementary condition:
𝑉
𝑉
𝑉
̂𝑗𝑉 ) − 𝛽𝑗𝑗 𝑥𝑟𝑗
−𝑃𝑗𝑋 (𝑋𝑗𝑉 , 𝑿
− ∑𝑙≠𝑗 𝛽𝑙𝑗 𝑥𝑟𝑙
+ 𝜇3𝑟𝑗 = 0, ∀𝑟, 𝑗,
which can be expressed in matrix notation as:
𝑉
−𝑷 𝑋0 − 𝜷𝒙𝑉 𝒆 − 𝜷𝑇 𝒙𝑉𝑟 + 𝝁𝟑𝑟 = 𝟎, ∀𝑟,
𝑉
𝑉 𝑇
where 𝒙𝑉 is a matrix with ( 𝒙1𝑉 , … , 𝒙𝑉𝐾 ) and each vector 𝒙𝑉𝑘 = (𝑥𝑘1
, … , 𝑥𝑘𝐽
) . 𝒆 is a vector
𝑉
(1, … ,1)𝑇 with K elements. 𝑷 𝑋0 is a price vector with elements 𝑃𝑗
𝑋0𝑉
. 𝝁𝟑𝑟 is a vector of the
Lagrangian multiplier with elements 𝜇3𝑟𝑗 . If 𝒙𝑉∗ is the solution obtained from the first order
𝑉
̂𝑗𝑉∗ ) = 𝑃
condition, then we need to show that 𝑃𝑗𝑋 (𝑋𝑗𝑉∗ , 𝑿
𝑗
𝑋0𝑉
+ 𝛽𝑗𝑗 𝑋𝑗𝑉∗ + ∑𝑙≠𝑗 𝛽𝑗𝑙 𝑋𝑙𝑉∗ ≥ 0 for all
𝑉
𝑗. We express this equation in the matrix notation 𝑷 𝑋0 + 𝜷𝒙𝑉∗ 𝒆. Obviously, the first order
𝑉
𝑉
𝑋0
condition gives 𝑷 𝑋0 + 𝜷𝒙𝑉∗ 𝒆 + 𝜷𝑇 𝒙𝑉∗
> 0 and 𝜷 have non-negative
𝑟 = 𝝁𝟑𝑟 > 𝟎 if 𝑷
𝑉
𝑉
elements. This implies that it is necessary to set (∑𝑘 𝜆𝑟𝑘 𝑋𝑘𝑗
− 𝑥𝑟𝑗
) = 0 in terms of MiCP; the
upper bound of input level is characterized by the least value at the free disposability hull of
inputs, and the lower bound is the input level described by the free disposability hull of outputs
𝑉
𝑉
shown in Theorem 4.1. Because (∑𝑘 𝜆𝑟𝑘 𝑋𝑘𝑗
− 𝑥𝑟𝑗
) = 0, that is, for cost minimization, the
whole quantity of supply market would be minimized to reach a lower price at the inverse supply
function, a firm’s best strategy is to reduce its input level and to produce on the production
frontier.
□
Generalized profit model as revenue maximization case
In a special case of the revenue model, we assume that the output level directly follows the
variable input, namely, the level of variable input determines and controls the level of output. For
example, in the semiconductor manufacturing industry raw silicon wafers are released into the
production line to generate the actual die output. If the yield is 100%, the output level is a linear
function of the variable input level. Assuming a constant unit cost of the variable input, we
formulate the profit maximization model as:
∑𝑘 𝜆𝑟𝑘 𝑌𝑘𝑞 ≥ 𝑦𝑟𝑞 ∀𝑞;
𝐹
𝐹
∑𝑘 𝜆𝑟𝑘 𝑋𝑘𝑖
≤ 𝑋𝑟𝑖
∀𝑖;
|
𝑃𝐹𝑟∗ = max
,
|
𝑉
𝑋
𝑦𝑟𝑞
̂ 𝑞 )𝑦𝑟𝑞
∑𝑘 𝜆𝑟𝑘 = 1;
− ∑𝑞 𝑃𝑞 (𝑌𝑞 , 𝒀
{
}
𝜆𝑟𝑘 ≥ 0 ∀𝑘;
̂ 𝑞 )𝑦𝑟𝑞
∑𝑞 𝑃𝑞𝑌 (𝑌𝑞 , 𝒀
(A2)
𝑉
𝑉
̂ 𝑞 ) becomes a constant and presents a unit cost of variable input 𝑥𝑟𝑗
where 𝑃𝑞𝑋 (𝑌𝑞 , 𝒀
= 𝜌𝑦𝑟𝑞 , and
𝜌 is a coefficient to change the units to a linear function. Intuitively, model (A2) is quite similar
to formulation (4), the revenue maximization model. The profit function of firm 𝑟
𝑉
𝑉
̂ 𝑞 ) − 𝑃𝑞𝑋 (𝑌𝑞 , 𝒀
̂ 𝑞 )]𝑦𝑟𝑞 is a concave function because 𝑃𝑞𝑋 (𝑌𝑞 , 𝒀
̂ 𝑞 ) is a constant and
∑𝑞[𝑃𝑞𝑌 (𝑌𝑞 , 𝒀
𝑉
̂ 𝑞 )𝑦𝑟𝑞 is a linear function. Thus, a Nash equilibrium exists and is unique. See Sections
𝑃𝑞𝑋 (𝑌𝑞 , 𝒀
3 and 4 in the paper for the theorems proving existence and uniqueness.
Price Sensitivity and Returns to Scale
It is necessary to understand the relationship between the price sensitivity matrices 𝜶 and 𝜷 and
the returns to scale (RTS) properties of the Nash equilibrium benchmarks. To address RTS
properties, we must first identify the most productive scale size (MPSS). The production frontier
is characterized by three regions: constant returns to scale (CRS), increasing returns to scale
(IRS), and decreasing returns to scale (DRS). The MPSS can be identified for firm 𝑟’s input and
output mix using the input-oriented CRS DEA technique formulated in (A3). If the sum of
2
∑𝑘 𝜆𝐶𝑅𝑆
𝑟𝑘 = 1 in the input-oriented CRS DEA , then we can identify such observations as
operating at the MPSS (Banker, 1984):
min
𝜃𝑟 ,𝜆𝐶𝑅𝑆
𝑟𝑘
∑𝑘 𝜆𝐶𝑅𝑆
𝑟𝑘 𝑌𝑘𝑞 ≥ 𝑌𝑟𝑞 , ∀𝑞, 𝑟;
𝐹
𝐹
∑ 𝜆𝐶𝑅𝑆
𝑟𝑘 𝑋𝑘𝑖 ≤ 𝑋𝑟𝑖 , ∀𝑖, 𝑟;
∑𝑟 𝜃𝑟 || 𝑘 𝐶𝑅𝑆
.
𝑉
𝑉
∑𝑘 𝜆𝑟𝑘 𝑋𝑘𝑗
≤ 𝜃𝑟 𝑋𝑟𝑗
, ∀𝑗, 𝑟;
{
𝜆𝐶𝑅𝑆
𝑟𝑘 ≥ 0, ∀𝑘, 𝑟;
(A3)
}
Let 𝑘 𝑀𝑃𝑆𝑆 denote the set of observations having the MPSS property, one for each firm 𝑟 in the
𝑉∗
∗
dataset, and let 𝑦𝑟𝑞
and 𝑥𝑟𝑗
be the Nash equilibrium solutions obtained from MiCP (9). Using
these additional observations as the reference set, optimization problem (A4) can be used to
identify the returns to scale property for each production plan in the Nash solution.
∗
∑𝑘 𝑀𝑃𝑆𝑆 𝜆𝐶𝑅𝑆
𝑌
≥ 𝑦𝑟𝑞
, ∀𝑞, 𝑟;
𝑟𝑘 𝑀𝑃𝑆𝑆 𝑘 𝑀𝑃𝑆𝑆 𝑞
∑𝑟 𝜃𝑟
min
𝜃𝑟 ,𝜆𝐶𝑅𝑆
𝑀𝑃𝑆𝑆
𝑟𝑘
{
𝐶𝑅𝑆
𝐹
𝐹
, ∀𝑖, 𝑟;
| ∑𝑘 𝑀𝑃𝑆𝑆 𝜆𝑟𝑘 𝑀𝑃𝑆𝑆 𝑋𝑘 𝑀𝑃𝑆𝑆 𝑖 ≤ 𝑋𝑟𝑖
𝐶𝑅𝑆
𝑉
𝑉∗
, ∀𝑗, 𝑟;
|∑𝑘 𝑀𝑃𝑆𝑆 𝜆𝑟𝑘 𝑀𝑃𝑆𝑆 𝑋𝑘 𝑀𝑃𝑆𝑆 𝑗 ≤ 𝜃𝑟 𝑥𝑟𝑗
𝜆𝐶𝑅𝑆
≥ 0, ∀𝑘, 𝑟;
𝑟𝑘 𝑀𝑃𝑆𝑆
.
(A4)
}
For each Nash solution of firm 𝑟, if ∑𝑘 𝑀𝑃𝑆𝑆 𝜆𝐶𝑅𝑆∗
< 1, then firm 𝑟 operates under increasing
𝑟𝑘 𝑀𝑃𝑆𝑆
returns to scale; if ∑𝑘 𝑀𝑃𝑆𝑆 𝜆𝐶𝑅𝑆∗
> 1, then firm 𝑟 operates under decreasing returns to scale;
𝑟𝑘 𝑀𝑃𝑆𝑆
or, if ∑𝑘 𝑀𝑃𝑆𝑆 𝜆𝐶𝑅𝑆∗
= 1, then firm 𝑟 operates under constant returns to scale.3 The equation
𝑟𝑘 𝑀𝑃𝑆𝑆
∑𝑘 𝑀𝑃𝑆𝑆 𝜆𝐶𝑅𝑆∗
is termed the RTS index (RTSI).
𝑟𝑘 𝑀𝑃𝑆𝑆
Note there are potential multiple optimal solutions. See Zhu (2000) for additional details.
In (9) note that both inputs and outputs are defined as adjustable; thus, all Nash equilibria are located on the
production frontier. If this is not the case, for example if there are adjustment costs when changing input levels (Choi
et al., 2006), then some equilibria may not be on the frontier as shown in Fig.3. For these equilibria RTS are not
defined because RTS is a frontier property. In (A4), if 𝜃𝑟 is not equal to 1, then RTS is not defined for that
production possibility; see, for example, Seiford and Zhu (1999) or Ray (2010).
2
3
For a one-input one-output production process, Fig.A2 depicts the true production
function as a solid curve, the CRS estimated frontier as a straight dashed line, the VRS estimated
frontier as a piece-wise linear bold dashed line, and the MPSS as Point B. In particular, based on
Theorem 4.1, the Nash equilibrium generated from MiCP (9) should be located on the bold
dashed lines. 𝑋𝐴 and 𝑋𝐸 are the upper and lower bounds for the variable input level.
Y (Output)
CRS estimated frontier
True Production Function
MPSS
B
IRS
DRS
VRS estimated frontier
A
F
C
D
E
XE
XA
X (Input)
Fig.A2 Nash equilibrium on bold dashed lines
Corollary A1: Assume all input and output variables are normalized to eliminate unit
dependence, and the price of outputs dominates the price of inputs to ensure a positive marginal
profit. Given a production frontier including three portions: IRS, CRS, and DRS, the MiCP (9)
generates a Nash equilibrium solution that is characterized by DRS when the inverse demand and
supply functions are less sensitive, or the Nash equilibrium is characterized by IRS when the
inverse demand and supply functions are more sensitive.
Proof: Intuitively, for one-variable-input one-output production process, if the inverse demand
and supply functions are less sensitive, this is illustrated in a special case where both price
sensitivity matrix 𝜶 and 𝜷 are equal to zero; the profit function 𝑃𝐹 can be written as maximizing
𝑉
𝑃𝐹 = 𝑃𝑌0 𝑦 − 𝑃 𝑋0 𝑥 𝑉 . Let 𝑃𝐹 ∗ denote the optimal value of profit function. Thus, we can express
𝑉
the function 𝑦 =
𝑃𝐹 ∗ +𝑃 𝑋0 𝑥 𝑉
𝑃 𝑌0
𝑉
, where
𝑃 𝑋0
𝑃 𝑌0
indicates the slope of profit function. Given the price of
𝑉
outputs dominating the price of inputs, in a special case the slope
𝑃 𝑋0
𝑃 𝑌0
→ 0+ , the optimal solution
to the firm specific profit maximization problem will result in a profit line tangent to the
𝑉
production possibility set. Since
𝑃 𝑋0
𝑃 𝑌0
→ 0+ , a firm would like to generate a Nash solution on the
DRS frontier for profit maximization based on Theorems 3.3 and 4.1, i.e., in extreme case, the
input level of the Nash solution has to be on the upper bound defined by the least value of the
free disposability hull of inputs (see firm A in Fig.A2). DRS is associated with the insensitive
inverse demand and supply function. Therefore, the Nash equilibrium solution generated from
MiCP (9) presents the DRS with respect to MPSS and ∑𝑘 𝑀𝑃𝑆𝑆 𝜆𝐶𝑅𝑆∗
= ∑𝑘 𝜆𝐶𝑅𝑆∗
> ∑𝑘 𝜆∗𝑟𝑘 =
𝑟𝑘
𝑟𝑘 𝑀𝑃𝑆𝑆
1. Similarly, we can show that the Nash solutions present the IRS when more sensitive inverse
demand and supply functions occur, i.e., a profit function with larger slope. The result can be
extended to the multiple-input multiple-output case.
□
Extending the illustrative example in Section 2, we use formulation (A4) to identify the
RTS property of the Nash equilibrium solution shown in Table 3; DCs 3, 10, 12, 15, and 19 are
identified as operating at MPSS. Table A1 shows the RTS associated with the Nash solutions for
Cases 1 through 5 in Table 3. Based on corollary A1, the sensitivity of output and sensitivity of
input are the two oppositional forces in terms of scale. Case 1 represents a baseline and the Nash
solutions present CRS or DRS properties. The sensitivity parameter of the supply function in
Case 2 increases relative to Case 1, which encourages firms to hold back on the consumption of
inputs, i.e., more DCs operate at MPSS in Case 2. If we further increase the sensitivity parameter
of the supply function, then all DCs operate at MPSS in Case 3. Case 4 results in all firms
operating at MPSS or IRS, by increasing the sensitivity parameter of the demand function and
leaving the sensitivity parameter of the supply function parameter the same as in Case 1. Case 5
shows that all firms operate at IRS and on the weakly efficient portion of the frontier. This
demonstrates the concept of rational inefficiency.
Table A1 Returns to scale of Nash equilibrium
Case
1
𝑞1
𝑞2
2
𝑗1
𝑗2
𝑞1
𝑞2
3
𝑗1
𝑗2
𝑞1
𝑞2
4
𝑗1
𝑗2
0.05 0.02 0.05 0.02 0.05 0.02 0.05 0.02 0.05 0.02 0.05 0.02
𝜶 or 𝜷
0.02 0.04 0.02 0.04 0.02 0.04 0.02
Returns to scale
RTSI
1
𝑞1
𝑞2
1
5
𝑗1
𝑗2
0.02 0.05 0.02
𝑞1
1
𝑞2
RTSI
RTS
RTSI
RTS
𝑗2
0.02 0.05 0.02
0.02 0.04 0.02 10 0.02 0.04 0.02 0.04 0.02
RTS
𝑗1
1
0.02 0.04
RTSI
RTS
RTSI
RTS
DC1
1
C
1
C
1
C
1
C
0.104
I
DC2
1.5
D
1
C
1
C
0.916
I
0.104
I
DC3
1
C
1
C
1
C
0.685
I
0.104
I
DC4
1.167
D
1
C
1
C
0.937
I
0.104
I
DC5
1.333
D
1
C
1
C
0.873
I
0.104
I
DC6
1
C
1
C
1
C
1
C
0.104
I
DC7
1.556
D
1.084
D
1
C
1
C
0.104
I
DC8
1.167
D
1
C
1
C
0.937
I
0.104
I
DC9
1.167
D
1
C
1
C
0.937
I
0.104
I
DC10
1
C
1
C
1
C
1
C
0.104
I
DC11
1.333
D
1
C
1
C
0.873
I
0.104
I
DC12
1.556
D
1.084
D
1
C
1
C
0.104
I
DC13
1.5
D
1
C
1
C
0.916
I
0.104
I
DC14
1
C
1
C
1
C
1
C
0.104
I
DC15
1.556
D
1.084
D
1
C
1
C
0.104
I
DC16
1.167
D
1
C
1
C
0.937
I
0.104
I
DC17
1.333
D
1
C
1
C
0.873
I
0.104
I
DC18
1.556
D
1.084
D
1
C
1
C
0.104
I
DC19
1
C
1
C
1
C
1
C
0.104
I
DC20
1.556
D
1.084
D
1
C
1
C
0.104
I
Returns to Scale
CRS or DRS
CRS or DRS
CRS
CRS or IRS
* C, D and I indicate constant, decreasing, and increasing returns to scale respectively.
IRS
Allocative Efficiency and Directional Distance Function
The Nash equilibrium identified by using (9) is an economically efficient solution, as shown in
Theorem 4.1. Zofio and Prieto (2006) suggest choosing the direction in the direction distance
function (DDF) to move towards the economically efficient point. Lee (2014) suggests the
direction towards marginal profit maximization. We extend their suggestion to the case of
imperfectly competitive markets and suggest that each firm should select the direction for
improvement in the DDF to move towards its Nash equilibrium benchmark.
The DDF, as defined by Chambers et al. (1996; 1998), is the simultaneous contraction of
inputs and expansion of outputs:
⃗ 𝑇 (𝑿𝐹 , 𝑿𝑉 , 𝒀; 𝒈𝑥 𝑉 , 𝒈𝑦 ) = max{𝛿 ∈ ℛ: (𝑿𝐹 , 𝑿𝑉 + 𝛿𝒈𝑥 𝑉 , 𝒀 + 𝛿𝒈𝑦 ) ∈ 𝑇},
𝐷
(A5)
𝑉
where 𝛿 is the distance measure and 𝒈𝑥 , 𝒈𝑦 are the direction vectors for variable inputs and
outputs, respectively. Recall that, since we do not change the fixed inputs in the short run, no
direction is associated with them. We estimate the DDF for firm 𝑟 as:
∑𝑘 𝜆𝑘 𝑌𝑘𝑞 ≥ 𝑌𝑟𝑞 + 𝛿𝑟 𝑔𝑞𝑦 , ∀𝑞;
𝐹
𝐹
∑𝑘 𝜆𝑘 𝑋𝑘𝑖
≤ 𝑋𝑟𝑖
, ∀𝑖;
|
⃗𝐷 𝑇̃ (𝑿𝐹𝑟 , 𝑿𝑉𝑟 , 𝒀𝑟 ; 𝒈 , 𝒈𝑦 ) = max 𝛿𝑟 ∑
𝑉
𝑉
𝑥𝑉
.
𝑘 𝜆𝑘 𝑋𝑘𝑗 ≤ 𝑋𝑟𝑗 + 𝛿𝑟 𝑔𝑗 , ∀𝑗;
𝛿𝑟 ,𝜆𝑘
|
∑𝑘 𝜆𝑘 = 1;
{
𝜆𝑘 ≥ 0, ∀𝑘;
}
𝑥𝑉
(A6)
𝑉
Because the method for selecting a direction (𝒈𝑥 , 𝒈𝑦 ) is an open issue, the direction (−𝟏, 𝟏) is
usually chosen for simplicity. Alternatively, Frei and Harker (1999) determine the least-norm
projection from an inefficient firm to the frontier, but this direction is non-proportional and is not
unit-invariant. Färe et al. (2013) estimate an endogenous direction, but it is void of economic
meaning. Therefore, we propose that firms’ direction for improvement move towards the
allocatively efficient benchmark, identified by the Nash equilibrium. Thus, the direction is firmspecific and can be calculated by following the equation for firm 𝑟:
𝑉
𝑦
(𝒙𝑉∗ −𝑿𝑉 ,𝒚∗ −𝒀 )
(𝒈𝑟𝑥 , 𝒈𝑟 ) = ‖(𝒙𝑟𝑉∗−𝑿𝑟𝑉,𝒚𝑟∗ −𝒀𝑟 )‖,
𝑟
𝑟
𝑟
𝑟
(A7)
∗
𝑉
where 𝒙𝑉∗
𝑟 and 𝒚𝑟 are the benchmarks determined by the Nash equilibrium, 𝑿𝑟 and 𝒀𝑟 are the
vectors of the current variable input and output production, and ‖∙‖ is the Euclidean norm. This
𝑉
ratio imposes (𝒈𝑥 , 𝒈𝑦 ) is a unit vector.4
Extending the example in Table 3, Case 1, we calculate the direction of improvement
associated with this example, as shown in Table A2. The results indicate that, when trying to
maximize overall economic efficiency5 using formulation (8), it is not necessary to contract the
variable inputs and expand the outputs. To maintain higher price and profit maximization, firm 𝑟
may achieve economic efficiency by changing its mix to become allocatively efficient. However,
no firm takes a direction which increases all variable inputs and decreases all output levels as this
would lead to a loss in profit.
Table A2 Direction determination
𝑉
Direction (𝒈𝑥 , 𝒈𝑦 )
Case 1
DC1
DC2
DC3
DC4
DC5
DC6
DC7
DC8
DC9
DC10
DC11
DC12
DC13
DC14
DC15
DC16
DC17
DC18
DC19
DC20
𝑗1
𝑗2
𝑞1
𝑞2
0.0467
0.0697
0.0000
-0.0445
0.0547
0.0445
-0.0478
0.0118
0.0427
0.0000
0.0000
0.1010
0.0000
0.0305
0.0532
0.0617
-0.0503
0.0213
0.0405
0.0000
0.0467
0.0697
0.0000
0.0000
0.0710
0.0222
-0.0717
0.0118
0.0427
0.0000
0.0000
0.1010
0.0000
0.0914
0.0710
0.0309
-0.1006
0.0213
0.0607
0.0265
0.7000
0.6970
0.0000
0.9785
0.8516
0.9783
-0.2390
0.8865
0.5341
0.0000
0.5255
0.8416
0.9955
0.9753
0.8875
0.9878
0.0503
0.9575
0.5059
0.3440
-0.7111
-0.7103
-1.0000
0.2012
-0.5165
0.2012
-0.9672
-0.4625
-0.8433
-1.0000
-0.8508
-0.5210
-0.0948
0.1988
-0.4522
0.1396
-0.9924
-0.2867
-0.8595
-0.9386
The length of the directional vector influences the efficiency estimates in the DDF; the use of a unit vector has also
been used in Fare et al. (2013).
5
Economic efficiency is the product of allocative efficiency and technical efficiency, see, for example, Fried et al.
(2008).
4
On-Line Electronic Supplementary Material B: Proofs
Lemma 2.1: Define 𝑃𝑞𝑌 (𝑦𝑟𝑞 )𝑦𝑟𝑞 as a concave function of 𝑦𝑟𝑞 and assume that the inverse
demand function 𝑃𝑞𝑌 (𝑦𝑟𝑞 ) is a non-increasing. Thus, for 𝑌̂𝑟𝑞 > 0, 𝑃𝑞𝑌 (𝑦𝑟𝑞 + 𝑌̂𝑟𝑞 )𝑦𝑟𝑞 is a concave
function of 𝑦𝑟𝑞 for 𝑦𝑟𝑞 ≥ 0, where 𝑌̂𝑟𝑞 = ∑𝑘≠𝑟 𝑦𝑘𝑞 . Similarly, let 𝑃𝑖𝑋 (𝑥𝑟𝑖 + 𝑋̂𝑟𝑖 )𝑥𝑟𝑖 be a convex
function of 𝑥𝑟𝑖 for 𝑥𝑟𝑖 ≥ 0, where 𝑋̂𝑟𝑖 = ∑𝑘≠𝑟 𝑥𝑘𝑖 and 𝑃𝑖𝑋 (𝑥𝑟𝑖 ) is an inverse supply function.
Furthermore, if either 𝑃𝑞𝑌 (𝑦𝑟𝑞 ) is strictly decreasing or is strictly convex, then 𝑃𝑞𝑌 (𝑦𝑟𝑞 + 𝑌̂𝑟𝑞 )𝑦𝑟𝑞
is a strictly concave function on the non-negative 𝑦𝑟𝑞 ≥ 0 and ∑𝑞 𝑃𝑞𝑌 (𝑦𝑟𝑞 + 𝑌̂𝑟𝑞 )𝑦𝑟𝑞 −
∑𝑖 𝑃𝑖𝑋 (𝑥𝑟𝑖 + 𝑋̂𝑟𝑖 )𝑥𝑟𝑖 is concave on (𝑥𝑟𝑖 , 𝑦𝑟𝑞 ) ∈ 𝑇̃.
Proof: Murphy et al. (1982) prove the single output product case that when 𝑦𝑟 ≥ 0 and 𝑌̂𝑟 > 0,
the revenue function 𝑅𝑟 = 𝑃𝑌 (𝑦𝑟 + 𝑌̂𝑟 )𝑦𝑟 is a concave function of 𝑦𝑟 for 𝑦𝑟 ≥ 0 on the nonnegative real line since
𝜕2 𝑅𝑟
𝜕𝑦𝑟2
< 0. In our special case, as Murphy et al. proved in their Lemma 1,
the production possibility set (𝒙, 𝒚) is a convex set and the boundary is a piece-wise linear
concave function. Thus, 𝑃𝑞𝑌 (𝑦𝑟𝑞 + 𝑌̂𝑟𝑞 )𝑦𝑟𝑞 is a concave function of 𝑦𝑟𝑞 for 𝑦𝑟𝑞 ≥ 0 and
(𝑥𝑟𝑖 , 𝑦𝑟𝑞 ) ∈ 𝑇̃ since, given fixed input levels, firm 𝑟 can expand output only by increasing 𝑦𝑟𝑞 .
Similarly, we can prove a convex cost function 𝑃𝑖𝑋 (𝑥𝑟𝑖 + 𝑋̂𝑟𝑖 )𝑥𝑟𝑖 of 𝑖 𝑡ℎ input resource, and also a
concave profit function ∑𝑞 𝑃𝑞𝑌 (𝑦𝑟𝑞 + 𝑌̂𝑟𝑞 )𝑦𝑟𝑞 − ∑𝑖 𝑃𝑖𝑋 (𝑥𝑟𝑖 + 𝑋̂𝑟𝑖 )𝑥𝑟𝑖 .
□
Theorem 2.1: If the profit function of firm 𝑟 , 𝜃𝑟 (𝒙𝑟 , 𝒚𝑟 ) = ∑𝑞 𝑃𝑞𝑌 (𝑌𝑞 )𝑦𝑟𝑞 − ∑𝑖 𝑃𝑖𝑋 (𝑋𝑖 )𝑥𝑟𝑖 is
concave with respect to (𝑥𝑟𝑖 , 𝑦𝑟𝑞 ) and continuously differentiable almost everywhere, where
𝑌𝑞 = ∑𝑘 𝑦𝑘𝑞 and 𝑋𝑖 = ∑𝑘 𝑥𝑘𝑖 , then (𝒙∗ , 𝒚∗ ) ∈ 𝑇̃ is a Nash imperfectly competitive market
equilibrium if and only if it satisfies the set of VI
〈𝐹((𝒙∗ , 𝒚∗ )), (𝒙, 𝒚) − (𝒙∗ , 𝒚∗ )〉 ≥ 0, ∀(𝒙, 𝒚) ∈ 𝑇̃ . That is,
∑𝑘 𝐹𝑘 ((𝒙∗ , 𝒚∗ ))((𝒙𝑘 , 𝒚𝑘 ) − (𝒙∗𝑘 , 𝒚∗𝒌 )) ≥ 0, ∀(𝒙𝑘 , 𝒚𝑘 ) ∈ 𝑇̃,
where 𝐹𝑘 ((𝒙, 𝒚)) = (−∇𝒙𝑘 𝜃𝑘 (𝒙, 𝒚), −∇𝒚𝑘 𝜃𝑘 (𝒙, 𝒚)), ∇𝒙𝑘 𝜃𝑘 (𝒙, 𝒚) = (
∇𝒚𝑘 𝜃𝑘 (𝒙, 𝒚) = (
𝜕𝜃𝑘 (𝒙,𝒚)
𝜕𝑦𝑘1
,…,
𝜕𝜃𝑘 (𝒙,𝒚)
𝜕𝑥𝑘1
,…,
𝜕𝜃𝑘 (𝒙,𝒚)
𝜕𝑥𝑘𝐼
) and
𝜕𝜃𝑘 (𝒙,𝒚)
𝜕𝑦𝑘𝑄
).
Proof: First we focus on revenue function with a single output. If the revenue function 𝑃(𝑌)𝑦𝑟 is
concave with respect to 𝑦𝑟 and continuously differentiable almost everywhere, then (𝑋𝑖 , 𝑦 ∗ ) ∈ 𝑇̃
is a Nash-Cournot imperfectly competitive market equilibrium if and only if it satisfies the set of
VI 〈𝐹(𝒚∗ ), 𝑦 − 𝑦 ∗ 〉 ≥ 0, ∀(𝑋𝑘𝑖 , 𝑦𝑘 ) ∈ 𝑇̃ . That is, ∑𝑘 𝐹𝑘 (𝒚∗ )(𝑦𝑘 − 𝑦𝑘∗ ) ≥ 0 ∀(𝑋𝑘𝑖 , 𝑦𝑘 ) ∈ 𝑇̃;
𝐹𝑘 (𝒚∗ ) = −𝑃(𝑌 ∗ ) − 𝑦𝑘∗ 𝑃′ (𝑌 ∗ ) . Since the revenue function 𝑃(𝑌)𝑦𝑟 is a continuously
differentiable almost everywhere and concave with respect to 𝑦𝑟 , for a fixed 𝑟 , the Nash
equilibrium condition 𝑃(𝑦𝑟∗ , 𝑦̂𝑟∗ )𝑦𝑟∗ − 𝑃(𝑦𝑟 , 𝑦̂𝑟∗ )𝑦𝑟 ≥ 0 , ∀(𝑋𝑟𝑖 , 𝑦𝑟 ) ∈ 𝑇̃ is equivalent to the
variational inequality 𝐹𝑟 (𝒚∗ )(𝑦𝑟 − 𝑦𝑟∗ ) ≥ 0 , i.e., 〈𝐹𝑟 (𝒚∗ ), 𝑦𝑟 − 𝑦𝑟∗ 〉 ≥ 0 , ∀(𝑋𝑟𝑖 , 𝑦𝑟 ) ∈ 𝑇̃ . Then,
summing over all firms 𝑘 generates 〈𝐹(𝒚∗ ), 𝑦 − 𝑦 ∗ 〉 ≥ 0, ∀(𝑋𝑘𝑖 , 𝑦𝑘 ) ∈ 𝑇̃ . This result can be
extended to prove the VI of multi-output revenue function, the VI of multi-input cost function,
and then the VI of profit function.
□
Theorem 2.2: Consider an imperfectly competitive market with 𝐾 firms, an inverse demand
function 𝑃𝑌 (∙) that is strictly decreasing and continuously differentiable in 𝑦, and an inverse
supply function 𝑃 𝑋 (∙) that is strictly increasing and continuously differentiable in 𝑥 . Since
Lemma 2.1 shows that the profit function 𝜃𝑘 (𝑥𝑘 , 𝑦𝑘 ) is concave and the variables 𝑥𝑘 , 𝑦𝑘 ≥ 0,
then (𝒙∗ , 𝒚∗ ) = ((𝒙1∗ , 𝒚1∗ ), (𝒙∗2 , 𝒚∗2 ), … , (𝒙∗𝐾 , 𝒚∗𝐾 )) is a Nash equilibrium solution if and only if
∇𝒙𝑘 𝜃𝑘 (𝒙∗ , 𝒚∗ ) ≤ 0 and ∇𝒚𝑘 𝜃𝑘 (𝒙∗ , 𝒚∗ ) ≤ 0, ∀𝑘;
𝒙∗𝑘 [∇𝒙𝑘 𝜃𝑘 (𝒙∗ , 𝒚∗ )] = 0 and 𝒚∗𝑘 [∇𝒚𝑘 𝜃𝑘 (𝒙∗ , 𝒚∗ )] = 0, ∀𝑘,
where (𝒙∗𝑘 , 𝒚∗𝑘 ) ∈ 𝑇̃.
Proof: We derive the formulas above based on the KKT conditions. Note that the KKT
conditions are both necessary and sufficient conditions for a unique global optimum since the
model maximizes a strictly concave profit function over a convex polyhedral set (the production
possibility set). The detail of existence and uniqueness of a Nash equilibrium is addressed in
Section 4 of the paper.
□
Lemma 3.1: A Nash solution to MiCP problem (3) will satisfy 𝑦𝑟 ≥ 0 and 𝑃(𝑌) ≥ 0.
Proof: 𝑃(𝑌) − 𝛼𝑦𝑟 − 𝜇1𝑟 = 0, that is 𝑃(𝑌) ≥ 𝛼𝑦𝑟 since 𝜇1𝑟 ≥ 0. If 𝑦𝑟 ≥ 0, then 𝑃(𝑌) ≥ 0 and
the revenue function is non-negative. If 𝑦𝑟 ≤ 0 and 𝑃(𝑌) ≥ 0, a firm’s best strategy to maximize
the revenue function is to make 𝑦𝑟 = 0. The case 𝑦𝑟 ≤ 0 and 𝑃(𝑌) < 0 will not happen because
if (𝑌) < 0 , then there exists at least one firm generating 𝑦𝑘 > 0, 𝑘 ≠ 𝑟 such that the function
𝑃(𝑌) = 𝑃0 − 𝛼𝑌 < 0. However, to maximize its revenue, firm 𝑘 prefers to produce 𝑦𝑘 = 0. In
other words, 𝑃(𝑌) = 𝑃0 − 𝛼𝑌 ≥ 0 if 𝑦𝑟 ≤ 0. In addition, if 𝛼 is a large positive number, 𝑦𝑟 can
be very small but positive to ensure a positive revenue function. Thus, any solution to this MiCP
(3) model enforces that 𝑦𝑟 and 𝑃(𝑌) are non-negative.
□
Theorem 3.1: If 𝑃(𝑌) = 𝑃0 − 𝛼𝑌 ≥ 0 and α is a small enough positive parameter, then the Nash
equilibrium solution is for all firms to produce on the production frontier.
𝜕𝐿
Proof: In MiCP, 𝜕𝑦𝑟 = (𝑃(𝑌) − 𝛼𝑦𝑟 − 𝜇1𝑟 ) = 0, ∀𝑟; where α is small enough, then we have
𝑟
𝑃(𝑌) − 𝛼𝑦𝑟 = 𝜇1𝑟 ≥ 0. In the extreme case, 𝛼 = 0, then 𝑃(𝑌) = 𝑃0 = 𝜇1𝑟 > 0. By MiCP, 0 ≥
(𝑦𝑟 − ∑𝑘 𝜆𝑟𝑘 𝑌𝑘 ) ⊥ 𝜇1𝑟 > 0, ∀𝑟 , which gives 𝑦𝑟 − ∑𝑘 𝜆𝑟𝑘 𝑌𝑘 = 0 . Once again, a firm’s best
strategy is to produce on the production frontier.
□
Theorem 3.2: If 𝑃(𝑌) = 𝑃0 − 𝛼𝑌 ≥ 0 and α is a large enough positive parameter, then the
MiCP will lead to a benchmark output level with 𝑦𝑟 = 𝑦̅𝑟 close to zero, where 𝑦̅𝑟 defines a
truncated output level.
Proof: Since 𝑃0 − 𝛼𝑌 ≥ 0 from Lemma 3.1 and 𝑃0 is a constant, then ≤
𝑃0
𝛼
, meaning that a
larger 𝛼 will result in a smaller 𝑌. In the MiCP, 0 ≥ (𝑦𝑟 − ∑𝑘 𝜆𝑟𝑘 𝑌𝑘 ) ⊥ 𝜇1𝑟 ≥ 0, ∀𝑟. If 𝑌 is
small, then (𝑦𝑟 − ∑𝑘 𝜆𝑟𝑘 𝑌𝑘 ) < 0, i.e., 𝜇1𝑟 = 0. In other words, we can increase α until no firm
would choose to produce on the production frontier in a Nash equilibrium solution, and then all
𝜇1𝑟 = 0, ∀𝑟. Proving this results for a truncated benchmark output level requires us to show that,
if α increases, then 𝑦𝑟 decreases and approaches zero. Since 𝜇1𝑟 = 0 and we know 𝑦𝑟 ≥ 0 by
lemma 3.1, 𝑃(𝑌) − 𝛼𝑦𝑟 = 0 in the MiCP and 𝑦𝑟 =
𝑃(𝑌)
𝑃(𝑌)
𝛼
𝛼
. In addition, 𝑦𝑟 =
0
there are only two firms in the market, 𝑦1 =
(𝑃 ⁄𝛼)−𝑦2
2
and 𝑦2 =
0
(𝑃 ⁄𝛼)−𝑦1
2
=
𝑃 0 −𝛼 ∑𝑘≠𝑟 𝑦𝑘
2𝛼
. If
𝑃0
, then 𝑦1 = 𝑦2 = 3𝛼 .
𝑃0
This constant 3𝛼 identifies the truncation output level for production. If there are 𝐾 firms in the
market, 𝑦𝑟 =
𝑃 0 −𝛼 ∑𝑘≠𝑟 𝑦𝑘
2𝛼
and 𝑦𝑟 =
𝑃0
2
𝑃0
𝑃0
𝐾−1
𝐾−2
)−(𝐾−1)( )+(
)𝑦𝑟 +(
) ∑𝑘≠𝑟 𝑦𝑘
𝛼
2𝛼
2
2
(
2
𝑃0
𝐾−1
∑𝑘≠𝑟 𝑦𝑘 =
(2𝑦𝑟 − ( 𝛼 ) + (𝐾 − 1) (2𝛼) − (
𝐾−2
replace ∑𝑘≠𝑟 𝑦𝑘 in equation 𝑦𝑟 , thus 𝑦𝑟 =
2
2
5−𝐾
) 𝑦𝑟 ) = 𝐾−2 ((
𝑃 0 −𝛼 ∑𝑘≠𝑟 𝑦𝑘
2𝛼
=
2
𝑃0 (𝐾−3)𝑃0
−
𝛼
(𝐾−2)𝛼
5−𝐾
2+
𝐾−2
, then we obtain
𝑃0
) 𝑦𝑟 + (2𝛼) (𝐾 − 3)). We
𝑃0
= (𝐾+1)𝛼 . Therefore, for 𝐾
𝑃0
firms 𝑦𝑟 = (𝐾+1)𝛼 = 𝑦̅𝑟 , and this constant 𝑦̅𝑟 identifies the benchmark output level. As α goes to
𝑃0
infinity, 𝑦̅𝑟 = (𝐾+1)𝛼 → 0.
□
Theorem 3.3: If the price sensitivity matrix 𝜶 satisfies WDD but is not necessarily symmetric,
then the MiCP (6) generates (𝑋𝑟𝑖 , 𝑦𝑟𝑞 ) ∈ 𝑇̃ , where 𝑦𝑟𝑞 will approach the efficient frontier for
small enough values of 𝛼𝑞𝑞 ; 𝑦𝑟𝑞 = 𝑦̅𝑟𝑞 is the truncated benchmark output level that approaches
zero as 𝛼𝑞𝑞 approaches infinity.
Proof: This is similar to theorems 3.1 and 3.2. We know
𝜕𝐿𝑟
𝜕𝑦𝑟
̂ 𝑞 ) − 𝛼𝑞𝑞 𝑦𝑟𝑞 − ∑ℎ≠𝑞 𝛼ℎ𝑞 𝑦𝑟ℎ ≤ 𝜇1𝑟𝑞 , ∀𝑟. If the 𝛼𝑞𝑞 value is small enough and we
= 𝑃𝑞 (𝑌𝑞 , 𝒀
consider a special case 𝛼𝑞𝑞 = 0, and the 𝜶 matrix is diagonally dominant, then
̂ 𝑞 ) = 𝑃𝑞0 ≤ 𝜇1𝑟 . Referring to the MiCP, 0 ≥ (𝑦𝑟𝑞 − ∑𝑘 𝜆𝑟𝑘 𝑌𝑘𝑞 ) ⊥ 𝜇1𝑟𝑞 > 0 ,
0 < 𝑃𝑞 (𝑌𝑞 , 𝒀
∀𝑟, 𝑞, meaning 𝑦𝑟𝑞 − ∑𝑘 𝜆𝑟𝑘 𝑌𝑘𝑞 = 0, or a firm’s best strategy is to produce on the production
frontier except for the portion associated with positive slacks and dual variables equal to zero on
the output constraints since increasing output does not affect the price reduction.
̂ 𝑞 ) = 𝑃𝑞0 − 𝛼𝑞𝑞 𝑌𝑞 − ∑ℎ≠𝑞 𝛼𝑞ℎ 𝑌ℎ ≥ 0
On the other hand, if the 𝛼𝑞𝑞 value is large enough, 𝑃𝑞 (𝑌𝑞 , 𝒀
and 𝑃𝑞0 is a constant, then 𝑌𝑞 ≤
𝑃𝑞0 −∑ℎ≠𝑞 𝛼𝑞ℎ 𝑌ℎ
𝛼𝑞𝑞
. As 𝛼𝑞𝑞 becomes larger, 𝑌𝑞 approaches zero.
Referring to the MiCP, 0 ≥ (𝑦𝑟𝑞 − ∑𝑘 𝜆𝑟𝑘 𝑌𝑘𝑞 ) ⊥ 𝜇1𝑟𝑞 ≥ 0 , for all 𝑟, 𝑞 . If 𝑌𝑞 is small, then
(𝑦𝑟𝑞 − ∑𝑘 𝜆𝑟𝑘 𝑌𝑘𝑞 ) < 0 and 𝜇1𝑟𝑞 = 0. In other words, we can increase 𝛼𝑞𝑞 until no firm would
choose to produce on the production frontier in a Nash equilibrium solution, and then all 𝜇1𝑟𝑞 =
0, ∀𝑟, 𝑞. Now we show as 𝛼𝑞𝑞 increases, then the truncated output level 𝑦𝑟𝑞 becomes smaller and
̂ 𝑞 ) − 𝛼𝑞𝑞 𝑦𝑟𝑞 − ∑ℎ≠𝑞 𝛼ℎ𝑞 𝑦𝑟ℎ − 𝜇1𝑟𝑞 ≤ 0 , and we
approaches zero. Since we derive 𝑃𝑞 (𝑌𝑞 , 𝒀
̂ 𝑞 ) − 𝛼𝑞𝑞 𝑦𝑟𝑞 − ∑ℎ≠𝑞 𝛼ℎ𝑞 𝑦𝑟ℎ = 0 in MiCP (6) and
know 𝜇1𝑟𝑞 = 0 and 𝑦𝑟𝑞 ≥ 0, thus 𝑃𝑞 (𝑌𝑞 , 𝒀
𝑃𝑞0 −𝛼𝑞𝑞 ∑𝑘≠𝑟 𝑦𝑘𝑞 −∑ℎ≠𝑞 𝛼𝑞ℎ 𝑌ℎ −∑ℎ≠𝑞 𝛼ℎ𝑞 𝑦𝑟ℎ
0 ≤ 𝑦𝑟𝑞 =
𝑦𝑟1 =
2𝛼𝑞𝑞
𝑃10
−
2𝛼11
𝑦𝑟1 = (1 −
∑𝑘≠𝑟 𝑦𝑘1
2
−
𝛼12
2𝛼11
𝑌2 −
2 −1
𝛼12 𝛼21 +𝛼21
𝑃10
)
[(
4𝛼11 𝛼22
2𝛼11
𝛼21
2𝛼11
𝛼
. Considering a two-output products example,
𝑦𝑟2 and 𝑦𝑟2 =
𝑃20
1
) − (2
11 2𝛼22
− 2𝛼21
𝑃20
2𝛼22
−
∑𝑘≠𝑟 𝑦𝑘2
2
2
𝛼21
) ∑𝑘≠𝑟 𝑦𝑘1
11 𝛼22
− 4𝛼
−
𝛼21
𝑌
2𝛼22 1
−
𝛼
𝛼12
2𝛼22
𝑦𝑟1 ;
𝛼
− 2𝛼12 𝑌2 + 4𝛼21 ∑𝑘≠𝑟 𝑦𝑘2 ]
11
11
can
be derived. Furthermore, 𝑌2 = 𝑦𝑟2 + ∑𝑘≠𝑟 𝑦𝑘2 finally gives
𝑦𝑟1 = (1 −
2
2
2𝛼12 𝛼21 +𝛼12
+𝛼21
4𝛼11 𝛼22
−1
)
[(
𝑃10
2𝛼11
−
𝑃0
(𝛼12 +𝛼21 )𝑃20
4𝛼11 𝛼22
1
1
2
𝛼21
2
4𝛼11 𝛼22
)−( −
−
𝛼12 𝛼21
4𝛼11 𝛼22
) ∑𝑘≠𝑟 𝑦𝑘1 + (
𝛼12
4𝛼11
+
𝛼21
4𝛼11
−
𝛼12
2𝛼11
) ∑𝑘≠𝑟 𝑦𝑘2 ].
𝛼
Based on WDD, 𝑦𝑟1 ≈ (2𝛼1 ) − (2) ∑𝑘≠𝑟 𝑦𝑘1 + (4𝛼21 ) ∑𝑘≠𝑟 𝑦𝑘2 . This result shows that 𝑦𝑟1 is a
11
11
function of 𝑦𝑘1 and 𝑦𝑘2 , not a variable of index 𝑟. Thus, 𝑦𝑟1 is limited by a truncated level 𝑦̅𝑟1
for all firms, since for all firms 𝑟 the same equation applies as does 𝑦𝑟1 for revenue
maximization. Similar equation can be derived for 𝑦𝑟2 . In addition, as 𝛼11 approaches infinity,
𝑦𝑟1 ≈
− ∑𝑘≠𝑟 𝑦𝑘1
2
. That is, 𝑦𝑟1 = 𝑦̅𝑟1 equals to zero. We can extend this result to outputs of more
than two. Therefore, the truncation point approaches zero as 𝛼𝑞𝑞 becomes large.
□
Corollary 3.1: If the price sensitivity matrix 𝜶 satisfies the MDD property and 𝛼𝑞𝑞 ≫ 𝛼ℎℎ , 𝑞 ≠
ℎ, then the solution to the MiCP (6) will satisfy 𝑦𝑟𝑞 < 𝑦𝑟ℎ ∀𝑟, 𝑞.
Proof: Corollary 3.1. is proven in the proof of Theorem 3.3
□
Theorem 4.1: Given arbitrary price sensitivity matrices 𝜶 and 𝜷 that satisfy WDD, MiCP (9)
𝑉∗ ∗
𝐹
generates all allocatively efficient Nash solutions (𝑋𝑟𝑖
, 𝑥𝑟𝑗
, 𝑦𝑟𝑞 ) ∈ 𝑇̃. These solutions are on the
frontier including the weakly efficient frontier, but excluding the portion of the frontier
associated with positive slacks and dual variables equal to zero on the input constraints.
𝑉
Proof: Based on Theorem 3.3, if 𝑦𝑟𝑞 > 0, then 𝑥𝑟𝑗
> 0 based on the no free lunch axiom (Färe et
𝑉
𝑉
𝑉
̂𝑗𝑉 ) − 𝛽𝑗𝑗 𝑥𝑟𝑗
al., 1985). According to (A4) −𝑃𝑗𝑋 (𝑋𝑗𝑉 , 𝑿
− ∑𝑙≠𝑗 𝛽𝑙𝑗 𝑥𝑟𝑙
+ 𝜇3𝑟𝑗 = 0 , that is,
𝑉
𝑉
𝑉
̂𝑗𝑉 ) + 𝛽𝑗𝑗 𝑥𝑟𝑗
𝑃𝑗𝑋 (𝑋𝑗𝑉 , 𝑿
+ ∑𝑙≠𝑗 𝛽𝑙𝑗 𝑥𝑟𝑙
= 𝜇3𝑟𝑗 . Consider that 𝛽𝑗𝑙 ≥ 0, 𝑃𝑗
𝑋0𝑉
> 0 and the 𝜷 matrix is
𝑉
𝑉
diagonally dominant; then 𝜇3𝑟𝑗 > 0. In MiCP (9), 0 ≥ (∑𝑘 𝜆𝑟𝑘 𝑋𝑘𝑗
− 𝑥𝑟𝑗
) ⊥ 𝜇3𝑟𝑗 > 0, ∀𝑟, 𝑗,
𝑉
𝑉
which gives ∑𝑘 𝜆𝑟𝑘 𝑋𝑘𝑗
− 𝑥𝑟𝑗
= 0. Based on Theorem 3.3 we know that 𝜇1𝑟𝑞 is determined
based on the sensitivity matrix 𝜶, i.e., 𝑦𝑟𝑞 − ∑𝑘 𝜆𝑟𝑘 𝑌𝑘𝑞 ≤ 0. Thus, a firm’s best strategy is to
adjust its variable input and output levels moving towards the production frontier. The solution is
allocatively efficient because the Nash solution accounts for prices. Furthermore, the equation
𝑉
𝑉
∑𝑘 𝜆𝑟𝑘 𝑋𝑘𝑗
− 𝑥𝑟𝑗
= 0 implies that the slacks of the input constraints are equal to zero and the
feasible region of the Nash solution is the production possibility set 𝑇̃ excluding the region for
which the input levels is larger than an anchor points that have dual variables equal to zero on the
inputs’ constraints. This restriction on the production possibility set implies an upper bound of
𝑉∗ ∗
𝐹
adjustable input level. Therefore, all Nash equilibrium solutions (𝑋𝑟𝑖
, 𝑥𝑟𝑗
, 𝑦𝑟𝑞 ) belong to 𝑇̃
excluding with this restriction.
Theorem 4.2: MiCP (9) generates a Nash equilibrium solution (𝑿𝐹 , 𝒙𝑉∗ , 𝒚∗ ) ∈ 𝑇̃ .
□
Proof: If an equilibrium output vector exists and 𝑥𝑟𝑗 > 0, 𝑦𝑟𝑞 > 0 , it must satisfy the first order
condition of MiCP (9). The complementary condition gives the following first order condition on
the output side:
̂ 𝑞 ) − 𝛼𝑞𝑞 𝑦𝑟𝑞 − ∑ℎ≠𝑞 𝛼ℎ𝑞 𝑦𝑟ℎ − 𝜇1𝑟𝑞 = 0, ∀𝑟, 𝑞.
𝑃𝑞𝑌 (𝑌𝑞 , 𝒀
This condition can be expressed in matrix notation as:
𝑷𝑌0 − 𝜶𝒚𝒆 − 𝜶𝑇 𝒚𝑟 − 𝝁𝟏𝑟 = 𝟎, ∀𝑟,
where 𝒚 is a matrix with ( 𝒚1 , … , 𝒚𝐾 ) and each vector 𝒚𝑘 = (𝑦𝑘1 , … , 𝑦𝑘𝑄 )𝑇 . 𝒆 is a vector
𝑌
(1, … ,1)𝑇 with K elements. 𝑷𝑌0 is a price vector with elements 𝑃𝑞 0 . 𝝁𝟏𝑟 is a vector of the
Lagrangian multiplier with elements 𝜇1𝑟𝑞 . If 𝒚∗ is the solution obtained from the first order
̂ ∗𝑞 ) = 𝑃𝑞𝑌0 − 𝛼𝑞𝑞 𝑌𝑞∗ − ∑ℎ≠𝑞 𝛼𝑞ℎ 𝑌ℎ∗ ≥ 0 for all 𝑞.
condition, then we need to show that 𝑃𝑞𝑌 (𝑌𝑞∗ , 𝒀
This equation can be expressed in matrix notation as 𝑷𝑌0 − 𝜶𝒚∗ 𝒆. Obviously, the first order
condition gives 𝑷𝑌0 − 𝜶𝒚∗ 𝒆 = 𝜶𝑇 𝒚∗𝒓 + 𝝁𝟏𝑟 ≥ 𝟎 if 𝒚∗𝒓 ≥ 𝟎 for all 𝑟 by Lemma 3.1.
Similar to the first order condition on the variable input side
𝑉
−𝑷 𝑋0 − 𝜷𝒙𝑽 𝒆 − 𝜷𝑇 𝒙𝑽𝒓 + 𝝁𝟑𝑟 = 𝟎, ∀𝑟,
𝑉
𝑉
𝑉 𝑇
where 𝒙𝑉 is a matrix with (𝒙1𝑉 , … , 𝒙𝑉𝐾 ) and each vector 𝒙𝑉𝑘 = (𝑥𝑘1
, … , 𝑥𝑘𝐽
) . 𝑷 𝑋0 is a price vector
with elements 𝑃𝑗
𝑋0𝑉
. 𝝁𝟑𝑟 is a vector of the Lagrangian multiplier with elements 𝜇3𝑟𝑗 . If 𝒙𝑉∗ is the
solution obtained from the first order condition, then we need to show the equation
𝑉
̂𝑗𝑉∗ ) = 𝑃
𝑃𝑗𝑋 (𝑋𝑗𝑉∗ , 𝑿
𝑗
𝑋0𝑉
+ 𝛽𝑗𝑗 𝑋𝑗𝑉∗ + ∑𝑙≠𝑗 𝛽𝑗𝑙 𝑋𝑙𝑉∗ ≥ 0 for all 𝑗. This equation can be expressed in
𝑉
matrix notation as 𝑷 𝑋0 + 𝜷𝒙𝑉 𝒆 . Obviously, the first order condition derives the results
𝑉
−𝜷𝑇 𝒙𝑉𝒓 + 𝝁𝟑𝑟 = 𝑷 𝑋0 + 𝜷𝒙𝑉 𝒆 ≥ 𝟎 if 𝒙𝑉∗
𝒓 ≥ 𝟎 for all 𝑟 by the estimated production possibility
set 𝑇̃ describing a positive lower bound of input level. Therefore, if an equilibrium vector exists,
it must equal (𝒙𝑉∗ , 𝒚∗ ).
To show that (𝒙𝑉∗ , 𝒚∗ ) is indeed an equilibrium vector, for any non-negative vector
(𝑿𝐹 , 𝒙𝑉′ , 𝒚′) ∈ 𝑇̃, where (𝑿𝐹 , 𝒙𝑉′ , 𝒚′) ≠ (𝑿𝐹 , 𝒙𝑉∗ , 𝒚∗ ), we consider (𝑿𝐹 , 𝒙𝑉′ , 𝒚′) in which all the
elements are equal to (𝑿𝐹 , 𝒙𝑉∗ , 𝒚∗ ) except for some 𝒙𝑉𝑟 , 𝒚𝑟 columns. We need to show that
𝑉
𝑉′
′
̂ ′𝑞 )𝑦𝑟𝑞
̂𝑗𝑉′ )𝑥𝑟𝑗
∑ ∑ 𝑃𝑞𝑌 (𝑌𝑞′ , 𝒀
− ∑ ∑ 𝑃𝑗𝑋 (𝑋𝑗𝑉′ , 𝑿
𝑟
𝑞
𝑟
𝑗
𝑉
𝑉∗
∗
̂ ∗𝑞 )𝑦𝑟𝑞
̂𝑗𝑉∗ )𝑥𝑟𝑗
≤ ∑ ∑ 𝑃𝑞𝑌 (𝑌𝑞∗ , 𝒀
− ∑ ∑ 𝑃𝑗𝑋 (𝑋𝑗𝑉∗ , 𝑿
𝑟
𝑞
𝑟
𝑗
for all 𝑟. Since 𝑃𝐹 is a strictly concave function under concavity and differentiability almost
everywhere assumptions for the maximization problem, and (𝒙𝑉∗ , 𝒚∗ ) satisfies the first order
condition and the KKT condition, then (𝒙𝑉∗ , 𝒚∗ ) must be a global optimum, i.e., the
complementary condition provides a Nash equilibrium solution:
𝑉
𝑉
𝑉′
𝑉∗
′
∗
̂ ′𝑞 )𝑦𝑟𝑞
̂𝑗𝑉′ )𝑥𝑟𝑗
̂ ∗𝑞 )𝑦𝑟𝑞
̂𝑗𝑉∗ )𝑥𝑟𝑗
∑𝑟 ∑𝑞 𝑃𝑞𝑌 (𝑌𝑞′ , 𝒀
− ∑𝑟 ∑𝑗 𝑃𝑗𝑋 (𝑋𝑗𝑉′ , 𝑿
≤ ∑𝑟 ∑𝑞 𝑃𝑞𝑌 (𝑌𝑞∗ , 𝒀
− ∑𝑟 ∑𝑗 𝑃𝑗𝑋 (𝑋𝑗𝑉∗ , 𝑿
for all 𝑟 and (𝑿𝐹 , 𝒙𝑉′ , 𝒚′) ∈ 𝑇̃.
□
Theorem 4.3: If the profit function is a strictly concave function on (𝑿𝐹 , 𝒙𝑉 , 𝒚) ∈ 𝑇̃ , that is
continuous and differentiable almost everywhere and the price sensitivity matrices 𝜶 and 𝜷
satisfy the WDD property, then the Nash equilibrium solution found using MiCP (9) is unique if
a solution exists for the maximization problem.
Proof: To prove the uniqueness, let two vectors (𝑿𝐹 , 𝒙̇ 𝑉 , 𝒚̇ ) and (𝑿𝐹 , 𝒙𝑉∗ , 𝒚∗ ) ∈ 𝑇̃ be solutions
and (𝑿𝐹 , 𝒙̇ 𝑉 , 𝒚̇ ) ≠ (𝑿𝐹 , 𝒙𝑉∗ , 𝒚∗ ) satisfy the variational inequality:
′
𝐹 𝑉∗ ∗
𝐹 𝑉′ ′
̃
∑𝑘 𝐹𝑘 ((𝑿𝐹 , 𝒙𝑉∗ , 𝒚∗ ))𝑇 ∙ ((𝑿𝐹𝑘 , 𝒙𝑉′
𝑘 , 𝒚𝑘 ) − (𝑿𝑘 , 𝒙𝑘 , 𝒚𝑘 )) ≥ 0, ∀(𝑿𝑘 , 𝒙𝑘 , 𝒚𝑘 ) ∈ 𝑇 ;
(A8)
′
𝐹 𝑉
𝐹 𝑉′ ′
̃
∑𝑘 𝐹𝑘 ((𝑿𝐹 , 𝒙̇ 𝑉 , 𝒚̇ ))𝑇 ∙ ((𝑿𝐹𝑘 , 𝒙𝑉′
𝑘 , 𝒚𝑘 ) − (𝑿𝑘 , 𝒙̇ 𝑘 , 𝒚̇ 𝑘 )) ≥ 0, ∀(𝑿𝑘 , 𝒙𝑘 , 𝒚𝑘 ) ∈ 𝑇 .
(A9)
𝑉∗ ∗
𝑉′ ′
′
Substituting 𝒙̇ 𝑉𝑘 , 𝒚̇ 𝑘 for 𝒙𝑉′
𝑘 , 𝒚𝑘 in (A8) and 𝒙𝑘 , 𝒚𝑘 for 𝒙𝑘 , 𝒚𝑘 in (A9) and adding the resulting
inequalities gives
∗
∑𝑘(𝐹𝑘 ((𝑿𝐹 , 𝒙𝑉∗ , 𝒚∗ )) − 𝐹𝑘 ((𝑿𝐹 , 𝒙̇ 𝑉 , 𝒚̇ )))𝑇 ∙ ((𝑿𝐹𝑘 , 𝒙̇ 𝑉𝑘 , 𝒚̇ 𝑘 ) − (𝑿𝐹𝑘 , 𝒙𝑉∗
𝑘 , 𝒚𝑘 ) ≥0.
However, this inequality does not satisfy the definition of strict monotonicity.
Thus, 𝒙̇ 𝑉 = 𝒙𝑉∗ , 𝒚̇ = 𝒚∗ and the solution is unique.
□
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